Separate the combined inequality into two individual inequalities.

The given inequality is $-1 \le \frac{5-2x}{3} < 2x-1$
This can be split into two inequalities:

1. $-1 \le \frac{5-2x}{3}$
2. $\frac{5-2x}{3} < 2x-1$

$$-1 \le \frac{5-2x}{3}$$

Multiply both sides by 3:

$$-1 \times 3 \le 5-2x$$ $$-3 \le 5-2x$$

Subtract 5 from both sides:

$$-3 - 5 \le -2x$$ $$-8 \le -2x$$

Divide both sides by -2. Remember to reverse the inequality sign when dividing by a negative number:

$$\frac{-8}{-2} \ge x$$ $$4 \ge x$$

So, $x \le 4$.

$$\frac{5-2x}{3} < 2x-1$$

Multiply both sides by 3:

$$5-2x < 3(2x-1)$$ $$5-2x < 6x-3$$

Add $2x$ to both sides:

$$5 < 6x + 2x - 3$$ $$5 < 8x - 3$$

Add 3 to both sides:

$$5 + 3 < 8x$$ $$8 < 8x$$

Divide both sides by 8:

$$\frac{8}{8} < x$$ $$1 < x$$

So, $x > 1$.

From Step 2, we have $x \le 4$.
From Step 3, we have $x > 1$.
Combining these two, we get the combined inequality:

$$1 < x \le 4$$